EGLL to KORD, chances are you’re flying straight to the face of a 100 knots headwind from the jet stream. That’s not gonna be the smoothest ride, but without a little more information, I’m not exactly sure what can be offered, since ‘swaying side to side’ could mean various things depending on interpretation. Was it just normal turbulence? Was the AP unable to keep a course? Was the plane also porpoising? We don’t know.
If you want to share a video, that would help a lot. You can’t upload it directly, but you can upload it to YouTube then share the link here. That would help a lot.
One other question, what was your flight plan like? Was it detailed, Great Arc, style, or was it a straight shot? The shape of the Globe doesn’t easily lend itself to single-waypoint Transatlantic flights if that was the case.